Editing Angle (section)

===Vertical and {{vanchor|adjacent}} angle pairs===
[[File:Vertical Angles.svg|thumb|150px|right|Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles. [[Hatch_mark#Congruency_notation|Hatch marks]] are used here to show angle equality.]]
{{redirect-distinguish|Vertical angle|Zenith angle}}

When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other.
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| A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called ''vertical angles'' or ''opposite angles'' or ''vertically opposite angles''. They are abbreviated as ''vert. opp. ∠s''.<ref name="tb">{{harvnb|Wong|Wong|2009|pp=161–163}}</ref>
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The equality of vertically opposite angles is called the ''vertical angle theorem''. [[Eudemus of Rhodes]] attributed the proof to [[Thales|Thales of Miletus]].<ref>{{cite book|author=Euclid|author-link=Euclid|title=The Elements|title-link=Euclid's Elements}} Proposition I:13.</ref>{{sfn|Shute| Shirk|Porter|1960|pp=25–27}} The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note,{{sfn|Shute| Shirk|Porter|1960|pp=25–27}} when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as:
* All straight angles are equal.
* Equals added to equals are equal.
* Equals subtracted from equals are equal.

When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle ''A'' equals ''x'', the measure of angle ''C'' would be {{nowrap|180° − ''x''}}. Similarly, the measure of angle ''D'' would be {{nowrap|180° − ''x''}}. Both angle ''C'' and angle ''D'' have measures equal to {{nowrap|180° − ''x''}} and are congruent. Since angle ''B'' is supplementary to both angles ''C'' and ''D'', either of these angle measures may be used to determine the measure of Angle ''B''. Using the measure of either angle ''C'' or angle ''D'', we find the measure of angle ''B'' to be {{nowrap|1=180° − (180° − ''x'') = 180° − 180° + ''x'' = ''x''}}. Therefore, both angle ''A'' and angle ''B'' have measures equal to ''x'' and are equal in measure.

[[File:Adjacentangles.svg|right|thumb|225px|Angles ''A'' and ''B'' are adjacent.]]
| ''Adjacent angles'', often abbreviated as ''adj. ∠s'', are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called ''complementary'', ''supplementary'', and ''explementary'' angles (see ''{{section link|#Combining angle pairs}}'' below).
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A [[Transversal (geometry)|transversal]] is a line that intersects a pair of (often parallel) lines and is associated with ''exterior angles'', ''interior angles'', ''alternate exterior angles'', ''alternate interior angles'', ''corresponding angles'', and ''consecutive interior angles''.{{sfn|Jacobs|1974|p=255}}